Applied Math in the High SchoolsNovember 1, 1999
The author (right), a loyal presence at the MCM sessions at SIAM annual meetings, is shown here with Ben Fusaro of Florida State University, founder and long-time director of the Mathematical Contest in Modeling.
As a veteran judge and enthusiastic booster of the Mathematical Contest in Modeling, I have long been amazed at the ability of teams from certain high schools---most notably the North Carolina School of Science and Mathematics---to compete successfully with teams composed of outstanding college students. To learn how the feat is accomplished, and how the nation's most fortunate high school students now spend their time, I resolved to visit NCSSM and several other leading secondary schools. At each, I was encouraged to visit classes, meet teachers, talk to students, dine in cafeterias, thumb through textbooks, fumble with graphing calculators, and otherwise immerse myself in the modern high school environment.
The classrooms I visited bore little resemblance to those in the schools my friends and I attended during the 1950s. In addition to overhead projectors, computers, graphing calculators, and multicolored erector sets designed to facilitate 3D visualization, I found students free to leave their seats, to converse (quietly) among themselves, and to cooperate in the solution of assigned problems. To a fossil like myself, these classrooms seemed like circuses, with teachers acting as ringmasters. Yet few of the remarks I chanced to overhear involved hairstyles, rock groups, fast food, athletic prowess, or criminal activity. Most of the students seemed genuinely curious about the topics du jour, and went about their business as if engaged in a game of Clue. To my considerable surprise, there was no obvious show of disappointment when the answer turned out to resemble max(0, 7 - 11x), rather than "Colonel Mustard in the conservatory with a meat cleaver."
The Electronic Classroom
The schools on my itinerary belong to what I suspect is a tiny minority in which classroom computers gather no dust. As expected, I found those and other electronic teaching aids to be in constant and typically effective use. When functioning as intended, they furnish unmistakably better ways of presenting certain subjects. By no means, however, do the weapons of electronic instruction invariably function as intended.
I was secretly pleased to learn that even seasoned classroom teachers, fully accustomed to the vagaries of electronic media, encounter many of the same unwelcome surprises I always seem to stumble over in my own (rare and typically clumsy) attempts to exploit the obvious potential of the new technology. It is hard to escape the conclusion that the full power of the modern electronic classroom will remain largely untapped until instructors are provided with technical assistants to operate the often vexatious hardware, along with adequate rehearsal time to interact one-on-one with those assistants. After all, timing and coordination are essential ingredients in every dog and pony show.
Despite seemingly irrevocable commitments to electronic teaching aids, I did hear words of caution from individual instructors. The medium, I was told again and again, too often becomes the message. It is exceedingly difficult to motivate students to master basic algebra and trigonometry in the presence of increasingly powerful symbolic manipulators, graphics programs, and high-speed number crunchers.
Whereas the best students in the classes I visited---a minuscule sample, to be sure---seemed to be able to solve simple equations and simplify unnecessarily awkward expressions, a surprising number did not. The wealth of electronic teaching aids now on---or about to enter---the market will not be assimilated overnight into the secondary school curriculum.
During the MCM, teams are free to use computers, calculators, libraries, and even human sources of background information.
Four Strategies for Teaching Motivated Math Students
At Thomas Jefferson High School, in Alexandria, Virginia, I was introduced to the members of not one but two complete MCM teams, still in residence after participating as underclassmen in 1998. Each had long since completed the school's standard math sequence, and was being "kept busy" with courses in complex variables, differential equations, and statistics, along with advanced science courses. All seemed to have enjoyed the MCM experience, and expressed interest in participating again. Several expressed regret that the school's modeling course had been discontinued, due to the lack of an interested teacher. TJ is a "magnet school," established during the 1980s, to prepare students from the six Virginia counties nearest to Washington for careers in mathematics, science, and technology. It is a four-year school, with a varied curriculum and a broad range of extracurricular activities. Students ordinarily live at home and commute to school each day.
NCSSM, in contrast, offers only the last two years of high school, is legally obligated to admit students from every county in North Carolina, and requires all enrollees to reside in dormitories located on its pleasant suburban campus---a converted hospital on the northern outskirts of Durham. The math curriculum is heavily oriented toward "hands-on" data analysis, as advocated by H.0. Pollak, a long-time SIAM member who served on the school's original board of trustees and visited frequently. Incoming students ordinarily take a year of precalculus, followed by a year of calculus. Both are taught from books written by members of the NCSSM faculty, to reflect Pollak's problem-solving orientation.
Because a few students in each incoming class have already mastered beginning calculus and/or linear algebra, a variety of more advanced courses is available as well. Those currently available include statistics, discrete math, a second year of calculus, and a course in modeling nominally designed to prepare students for participation in the MCM. As at Thomas Jefferson, I met several students who had already competed once in the contest and were looking forward to doing so again.
The other two schools I visited were chosen mainly for their compatibility with my own rather limited travel plans. The Baltimore Polytechnic Institute was founded in 1883, as one of the city's two honors high schools for men, and retained that status until the mid-1970s, when it embraced coeducation. The fraction of female enrollees has since risen to 60%, and counting. Poly offers an A course and a B course, both of which prepare students to enter accredited engineering schools. Indeed, "the Poly A course" is locally famous for preparing students to enter even the most demanding engineering schools as sophomores.
Students in the A course take a semester each of trigonometry and probability/statistics in their sophomore year, along with the traditional year of plane geometry. This enables them to complete three semesters of calculus before graduation and, in many cases, to earn exemption from freshman calculus in college. (Without some exemptions, the proliferation of requirements has rendered it all but impossible to graduate from most engineering schools within the advertised four years.) For particularly well prepared students, Poly offers single-semester senior electives in discrete math and ordinary differential equations, piped in via fiber-optic cable from nearby Towson University.
Finally, Groton School is a highly selective private school in central Massachusetts, founded in 1884, and perhaps best known for its contributions to the New Deal---FDR graduated in 1900, Dean Acheson in 1908, Averell Harriman in 1911, and Sumner Welles in 1913. Although the curriculum has never emphasized math and science---two years of Latin, two of a modern language, and three of history are still required for graduation---the school has always prided itself on the rigor of its instruction in both. Juniors and seniors have performed creditably on Advanced Placement exams, and, over the years, a small number of graduates (including this reporter) have gravitated into scientific or technical careers.
At the Bottom Line . . .
All such schools are challenged by a trickle of particularly precocious students, of whom the most advanced are ready at the age of 16 for upper-division course work in any research university. The challenge is taken seriously, if confronted in different ways, by all the schools considered in this article. While some provide upper-division course work, others prefer to expand their curricula horizontally by introducing ancillary materials that might otherwise be overlooked. Various topics in discrete math---such as graph, number, and combinatorial theory---are proving ideally suited to this purpose. Students with an orientation toward physics might react well to David Acheson's book From Calculus to Chaos (see review, SIAM News, July/August 1999).
Students in such schools encounter almost as much mathematics outside their formal math classes as within them. One young man at Thomas Jefferson told me about his course in "computational physics," which had just finished a unit on orbital computations. Another was learning about optimization and operations research, while a third offered an exceedingly lucid description of a computer program he was learning to use in his advanced chemistry class.
Called GAUSSIAN, the package employs user-supplied information concerning bond lengths, interatomic angles, and so forth to analyze chemical reactions between reactants as complex as proteins containing scores or even hundreds of atoms. To that end, the program accesses a library of quantum mechanical subroutines. Simple reactions can be analyzed in minutes, while complex ones (like those involving large proteins) can require hours or even days to complete. GAUSSIAN, originally released in 1970, was developed by John A. Pople, a co-recipient of the Nobel Prize in Chemistry. More than 10,000 scientists are said to use the latest version of it.
In the course of learning to use the sort of off-the-shelf programs upon which applied math thrives, students acquire knowledge about the actual and potential uses of mathematics. I suspect that such experiences, which closely resemble the ones accumulated in an industrial laboratory or research center, are a significant advantage in a contest like the MCM, and go a long way toward explaining the success enjoyed by participating high school teams. Even brief exposure to the uses of these sophisticated commercial packages seems to impart a feel for what can and cannot be accomplished computationally.
I was particularly struck by the extent to which graph theory is making its way into the high school classroom. Students seem almost as familiar with Euler's formula as they do with the Pythagorean theorem. At NCSSM I attended a session on discrete math in which the students were making their way through the opening chapter of a beginning undergraduate-level text (David W. Farmer and Theodore B. Stanford, Knots and Surfaces: A Guide to Discovering Mathematics, American Mathematical Society, Providence, Rhode Island, 1995) and had been asked---by way of homework---to determine all possible graphs on five vertices. The exercise is trivial in the cases in which only a few edges are either missing or present; it becomes trickier as the numbers in the two categories near equality, requiring separation of the candidates into equivalence classes under isomorphism. The difficulty of the problem---at least on this occasion---elevated the tenor of debate to what seemed a heady level of abstraction.
The emphasis on graph theory was most evident at Groton, where it constitutes a large part of 10th-grade geometry. At first, I was appalled that an 18th-century upstart could be allowed to usurp even a small part of Euclid's traditional place in the curriculum. De-emphasize ruler-and-compass constructions? Forego two-column proofs? On reflection, however, I began to appreciate the method in Groton's madness. Euclidean geometry owes its prominent position in the secondary curriculum to the fact that its numerous theorems can be deduced-with apparent rigor---from a few relatively self-evident axioms. But the theorems of elementary graph theory can be deduced with even greater rigor from far less dubious axioms. Moreover, they lead more directly to modern applications, especially in OR and computer science. Why then should graph theory, which is equally visual, not supplant plane geometry as a training ground for deductive logic?
Elsewhere in Groton's decidedly nontraditional curriculum, there is a heavy emphasis on dynamic modeling, through functional iteration and finite-difference equations. Computers are used extensively, as are graphing calculators and other classroom aids. In one class, I saw seniors exploring contractive linear transformations in the plane, leading ultimately---via Michael Barnsley's collage theorem and computer graphics packages based on it---to discussions of algorithmic complexity and data compression. For all this Jon Choate, chair of the school's math department, has established a national reputation as a curriculum innovator. Choate's central thesis (which he stands ready to defend at any hour of day or night) is that high school students have plenty of time to learn about calculus, differential equations, and modern algebra in college . . . if they want to! Why not use the high school years to acquaint them with as many as possible of the important subjects that can be understood without mastering all that machinery?
The bottom line, if there is one, is that I still don't know how teams from magnet high schools remain so competitive in the MCM, year after year. But I have experienced electronic learning, as used by those who appear to use it best, and identified a few other factors that may contribute to the success of teams from these elite secondary schools. I only wish that more teenagers, both foreign and domestic, could attend institutions in which morale is as high---and the atmosphere as conducive to learning---as it is in the four schools I visited.
James Case is an independent consultant who lives in Baltimore, Maryland.